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Statistics

Statistics

Histograms

Histogram- Range Table Diff Cw Paper 1 + 2 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Drawing a histogram
  4. Reading a histogram
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 9.7: histograms
  • 9.7Calculate frequency density and draw a histogram with unequal class widths.
  • 9.7Read frequencies back from a histogram: area = frequency.
  • 9.7Estimate how many values lie in part of a class.

Before you start: Grouped data.

Formula bank

Frequency densityLearn it\(\text{frequency density}=\dfrac{\text{frequency}}{\text{class width}}\)the HEIGHT of each bar.Use it when drawing a histogram.
FrequencyLearn it\(\text{frequency}=\text{frequency density}\times\text{class width}\)the AREA of each bar.Use it when reading a histogram.
Part of a classLearn it\(\dfrac{\text{part width}}{\text{class width}}\times\text{frequency}\)assumes values spread evenly.Use it when 'estimate how many between…'.

Drawing a histogram

In a histogram the bars touch, the horizontal axis is a continuous scale, and the vertical axis is frequency density. The AREA of a bar shows the frequency, so a wide class doesn't look unfairly big.

Height h (cm)FrequencyClass widthFrequency density
\(0<h\le10\)8100.8
\(10<h\le15\)1252.4
\(15<h\le20\)1553
\(20<h\le30\)10101
\(30<h\le50\)6200.3

General methodHow to draw any histogram

  1. Work out each class width (upper − lower).
  2. Frequency density = frequency ÷ class width; add a column.
  3. Label the vertical axis 'Frequency density' and choose a scale.
  4. Draw each bar from its lower to its upper bound, as tall as its frequency density. No gaps.
Histogram with unequal class widths: height = frequency density10203040500.51.01.52.02.53.03.5hfd
The 30-50 class is wide, so its bar is low: area 0.3 × 20 = 6.

Check with areas: the first bar is 10 wide and 0.8 tall, area 8 = its frequency. If all classes have the same width, the histogram looks just like a bar chart with no gaps.

Try it yourself

1
A class \(50<m\le60\) has frequency 24. Frequency density?
Show answer

Answer 2.4

Reading a histogram

General methodHow to read frequencies from a histogram

  1. Read the bar's height (frequency density) on the scale.
  2. Frequency = height × class width.
  3. For part of a class, use only the part of the width you need.

Example Estimate how many heights are between 12 cm and 25 cm.

Parts of three bars: 12-15 (3 of 5 cm), all of 15-20, 20-25 (5 of 10 cm).

  1. \(2.4\times3=7.2\)
  2. \(3\times5=15\)
  3. \(1\times5=5\)
  4. \(7.2+15+5=27.2\)

Answer about 27

Try it yourself

1
A bar 5 wide has frequency density 1.6. Frequency?
Show answer

Answer 8

2
A bar for a class 20 wide has frequency density 0.45. Frequency?
Show answer

Answer 9

Ways they test it

Type 1 Draw

"Draw a histogram to show the information." (3 marks)

Spot it a grouped table with unequal widths.

  1. Frequency density column.
  2. Bars to scale.

Type 2 Complete the table from the histogram

"Complete the frequency table." (2 marks)

Spot it some bars drawn, some frequencies missing.

  1. Height × width.

For example bar 30-50, height 0.3

  1. \(0.3\times20\)

Answer 6

Type 3 Find the scale

"There are 8 values in the first class. Label the vertical scale." (2 marks)

Spot it no numbers on the vertical axis.

  1. Frequency density = 8 ÷ width.
  2. That fixes the scale.

Type 4 Part of a class

"Estimate the number of heights between 12 and 25 cm." (3 marks)

Spot it bounds inside classes.

  1. Fraction of each class.
  2. Add.

Mistakes to avoid

  • ✗ Bar height = frequency✓ height = frequency densityOnly for equal widths does this not matter.
  • ✗ Gaps between bars✓ bars touchThe scale is continuous.
  • ✗ Reading the height as the frequency✓ height × width
  • ✗ Vertical axis labelled 'Frequency'✓ 'Frequency density'It costs a mark.

Summary

  • Frequency density = frequency ÷ class width = bar height.
  • Area of a bar = frequency.
  • Part of a class: fraction of its width × its frequency.

If you see …

  • unequal class widthsfrequency density.
  • 'how many' from a histogramheight × width.

Practice

Practice: easy, medium, hard

Easy

1
Frequency density for \(0<h\le10\), frequency 8?
Show answer

Answer 0.8

2
Frequency density for \(10<h\le15\), frequency 12?
Show answer

Answer 2.4

3
Class width of \(30<h\le50\)?
Show answer

Answer 20

Medium

1
Bar 30-50 has height 0.3. Frequency?
Guide me

What is it asking? a frequency

What do we know? width 20

Plan \(0.3\times20\)

Show working
  1. 6

Answer 6

2
How many heights are 15 cm or less (from the histogram)?
Guide me

What is it asking? a count

What do we know? two bars

Plan \(0.8\times10+2.4\times5\)

Show working
  1. 20

Answer 20

3
A bar 5 wide has height 1.6. Frequency?
Guide me

What is it asking? a frequency

What do we know? height × width

Plan \(1.6\times5\)

Show working
  1. 8

Answer 8

Hard

1
Estimate how many heights lie between 12 cm and 25 cm (1 d.p.).
Guide me

What is it asking? a count

What do we know? 3 cm of the 12-15 bar, all of 15-20, 5 cm of 20-30

Plan 7.2 + 15 + 5

Show working
  1. 27.2

Answer 27.2

PartAnswer / workingMarksGuidance
\(2.4\times3\)M1
\(1\times5\)M1
27.2 (accept 27)A1
Total: 3 marks
2
Estimate how many heights lie between 20 cm and 25 cm.
Guide me

What is it asking? a count

What do we know? half of the 20-30 bar

Plan \(1\times5\)

Show working
  1. 5

Answer 5

PartAnswer / workingMarksGuidance
5B1
Total: 1 mark
3
A class 20 wide has frequency density 0.45. Frequency?
Guide me

What is it asking? a frequency

What do we know? area

Plan \(0.45\times20\)

Show working
  1. 9

Answer 9

PartAnswer / workingMarksGuidance
9B1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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